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The '''quantum Boltzmann equation,''' also known as the '''Uehling–Uhlenbeck equation''',<ref>{{cite journal |last1=Filbet |first1=Francis |last2=Hu |first2=Jingwei |last3=Jin |first3=Shi |title=A Numerical Scheme for the Quantum Boltzmann Equation Efficient in the Fluid Regime |journal=Esaim: M2An |year=2012 |volume=46 |issue=2 |pages=443–463|doi=10.1051/m2an/2011051|arxiv=1009.3352}}</ref><ref>{{cite book |last1=Bao |first1=Weizhu |last2=Markowich |first2=Peter |last3=Pareschi |first3=Lorenzo|chapter=Quantum kinetic theory: Modelling and numerics for Bose-Einstein condensation |title=Modeling and Computational Methods for Kinetic Equations |series=Modeling and Simulation in Science, Engineering and Technology |year=2004|pages=287–320|doi=10.1007/978-0-8176-8200-2_10|isbn=978-1-4612-6487-3 }}</ref> is the [[Physics:Quantum mechanics|quantum mechanical]] modification of the [[Boltzmann equation]], which gives the nonequilibrium [[Time evolution|time evolution]] of a gas of quantum-mechanically interacting particles. Typically, the quantum Boltzmann equation is given as only the “collision term” of the full Boltzmann equation, giving the change of the momentum distribution of a locally homogeneous gas, but not the drift and diffusion in space. It was originally formulated by [[Biography:Lothar Wolfgang Nordheim|L.W. Nordheim]] (1928),<ref>{{Cite journal|last1=Nordhiem|first1=L. W.|last2=Fowler|first2=Ralph Howard|date=1928-07-02|title=On the kinetic method in the new statistics and application in the electron theory of conductivity|journal=Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character|volume=119|issue=783|pages=689–698|doi=10.1098/rspa.1928.0126|bibcode=1928RSPSA.119..689N|doi-access=free}}</ref> and by and E. A. Uehling and [[Biography:George Uhlenbeck|George Uhlenbeck]] (1933).<ref>{{Cite journal|last1=Uehling|first1=E. A.|last2=Uhlenbeck|first2=G. E.|date=1933-04-01|title=Transport Phenomena in Einstein-Bose and Fermi-Dirac Gases. I|url=https://link.aps.org/doi/10.1103/PhysRev.43.552|journal=Physical Review|language=en|volume=43|issue=7|pages=552–561|doi=10.1103/PhysRev.43.552|bibcode=1933PhRv...43..552U|issn=0031-899X|url-access=subscription}}</ref>

In full generality (including the p-space and x-space drift terms, which are often neglected) the equation is represented analogously to the Boltzmann equation.
<math display="block">
\left[\frac{\partial}{\partial t} + \mathbf{v} \cdot \nabla_x + \mathbf{F} \cdot \nabla_p \right] f(\mathbf{x},\mathbf{p},t) = \mathcal{Q}[f](\mathbf{x},\mathbf{p})
</math>

where <math>\mathbf{F}</math> represents an externally applied potential acting on the gas' p-space distribution and <math>\mathcal{Q}</math> is the collision operator, accounting for the interactions between the gas particles. The quantum mechanics must be represented in the exact form of <math>\mathcal{Q}</math>, which depends on the physics of the system to be modeled.<ref>{{cite journal |last1=Filbert |first1=Francis |last2=Hu |first2=Jingwei |last3=Jin |first3=Shi |title=A Numerical Scheme for the Quantum Boltzmann Equation Efficient in the Fluid Regime |journal=Esaim: M2An |year=2012 |volume=46 |issue=2 |pages=443–463|doi=10.1051/m2an/2011051|arxiv=1009.3352}}</ref>

The quantum Boltzmann equation gives irreversible behavior, and therefore an [[Physics:Entropy (arrow of time)|arrow of time]]; that is, after a long enough time it gives an equilibrium distribution which no longer changes. Although quantum mechanics is microscopically time-reversible, the quantum Boltzmann equation gives irreversible behavior because phase information is discarded<ref>{{cite journal |last1=Snoke |first1=D.W. |last2=Liu |first2=G. |last3=Girvin |first3=S.M. |title=The basis of the Second Law of thermodynamics in quantum field theory |journal=[[Physics:Annals of Physics|Annals of Physics]] |year=2012 |volume=327 |issue=7 |pages=1825–1851 |bibcode=2012AnPhy.327.1825S |doi=10.1016/j.aop.2011.12.016|arxiv=1112.3009 |s2cid=118666925 }}</ref> only the average occupation number of the quantum states is kept. The solution of the quantum Boltzmann equation is therefore a good approximation to the exact behavior of the system on time scales short compared to the [[Poincaré recurrence theorem|Poincaré recurrence time]], which is usually not a severe limitation, because the Poincaré recurrence time can be many times the [[Astronomy:Age of the universe|age of the universe]] even in small systems.

The quantum Boltzmann equation has been verified by direct comparison to time-resolved experimental measurements, and in general has found much use in [[Physics:Semiconductor|semiconductor]] optics.<ref>{{cite journal |last1=Snoke |first1=D.W. |title=The quantum Boltzmann equation in semiconductor physics |journal=[[Physics:Annalen der Physik|Annalen der Physik]] |year=2011 |volume=523 |issue=1–2 |pages=87–100 |bibcode=2011AnP...523...87S |doi=10.1002/andp.201000102|arxiv=1011.3849 |s2cid=119250989 }}</ref> For example, the energy distribution of a gas of [[Physics:Exciton|exciton]]s as a function of time (in picoseconds), measured using a [[Engineering:Streak camera|streak camera]], has been shown<ref>{{cite journal |last1=Snoke |first1=D. W. |last2=Braun |first2=D. |last3=Cardona |first3=M. |title=Carrier thermalization in Cu<sub>2</sub>O: Phonon emission by excitons |journal=[[Physics:Physical Review B|Physical Review B]] |year=1991 |volume=44 |issue=7 |pages=2991–3000 |bibcode=1991PhRvB..44.2991S |doi=10.1103/PhysRevB.44.2991|pmid=9999890 }}</ref> to approach an equilibrium [[Maxwell–Boltzmann distribution|Maxwell-Boltzmann distribution]].

== Application to semiconductor physics ==
A typical model of a semiconductor may be built on the assumptions that:
# The electron distribution is spatially homogeneous to a reasonable approximation (so all x-dependence may be suppressed)
# The external potential is a function only of position and isotropic in p-space, and so <math>\mathbf{F}</math> may be set to zero without losing any further generality
# The gas is sufficiently dilute that three-body interactions between electrons may be ignored.

Considering the exchange of momentum <math>\mathbf{q}</math> between electrons with initial momenta <math>\mathbf{k}</math> and <math>\mathbf{k_1}</math>, it is possible to derive the expression
<math display="block">
\begin{aligned}
\mathcal{Q}[f](\mathbf{k})
&= \frac{-2}{\hbar (2\pi)^5}\int d\mathbf{q} \int d\mathbf{k_1}
|\hat{v}(\mathbf{q})|^2 \\
&\quad\times
\delta\!\left(\frac{\hbar^2}{2m}(|\mathbf{k-q}|^2+|\mathbf{k_1+q}|^2-\mathbf{k}_1^2-\mathbf{k}^2)\right)
\\
&\quad\times \left[f_{\mathbf{k}} f_{\mathbf{k_1}} (1-f_{\mathbf{k-q}})(1-f_{\mathbf{k_1+q}})
- f_{\mathbf{k-q}} f_{\mathbf{k_1+q}} (1-f_{\mathbf{k}})(1-f_{\mathbf{k_1}})\right]
\end{aligned}
</math>

== References ==
{{reflist}}

[[Category:Statistical mechanics]]

{{Sourceattribution|Quantum Boltzmann equation}}

Physics:Quantum Boltzmann equation - Revision history

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